Solve 116=x+15*6.3+10*30.8/x

$$116 = x + \frac { 15 \times 6.3 + 10 \times 30.8 } { x }$$

Solve for x

$$x=\frac{\sqrt{11846}}{2}+58\approx 112.419665563$$

$$x=-\frac{\sqrt{11846}}{2}+58\approx 3.580334437$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x.

$$116x=xx+15\times 6.3+10\times 30.8$$

Multiply x and x to get x^{2}.

$$116x=x^{2}+15\times 6.3+10\times 30.8$$

Multiply 15 and 6.3 to get 94.5.

$$116x=x^{2}+94.5+10\times 30.8$$

Multiply 10 and 30.8 to get 308.

$$116x=x^{2}+94.5+308$$

Add 94.5 and 308 to get 402.5.

$$116x=x^{2}+402.5$$

Subtract x^{2} from both sides.

$$116x-x^{2}=402.5$$

Subtract 402.5 from both sides.

$$116x-x^{2}-402.5=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$-x^{2}+116x-402.5=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 116 for b, and -402.5 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-116±\sqrt{116^{2}-4\left(-1\right)\left(-402.5\right)}}{2\left(-1\right)}$$

Square 116.

$$x=\frac{-116±\sqrt{13456-4\left(-1\right)\left(-402.5\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-116±\sqrt{13456+4\left(-402.5\right)}}{2\left(-1\right)}$$

Multiply 4 times -402.5.

$$x=\frac{-116±\sqrt{13456-1610}}{2\left(-1\right)}$$

Add 13456 to -1610.

$$x=\frac{-116±\sqrt{11846}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-116±\sqrt{11846}}{-2}$$

Now solve the equation x=\frac{-116±\sqrt{11846}}{-2} when ± is plus. Add -116 to \sqrt{11846}.

$$x=\frac{\sqrt{11846}-116}{-2}$$

Divide -116+\sqrt{11846} by -2.

$$x=-\frac{\sqrt{11846}}{2}+58$$

Now solve the equation x=\frac{-116±\sqrt{11846}}{-2} when ± is minus. Subtract \sqrt{11846} from -116.

$$x=\frac{-\sqrt{11846}-116}{-2}$$

Divide -116-\sqrt{11846} by -2.

$$x=\frac{\sqrt{11846}}{2}+58$$

The equation is now solved.

$$x=-\frac{\sqrt{11846}}{2}+58 x=\frac{\sqrt{11846}}{2}+58$$

Steps for Completing the Square

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x.

$$116x=xx+15\times 6.3+10\times 30.8$$

Multiply x and x to get x^{2}.

$$116x=x^{2}+15\times 6.3+10\times 30.8$$

Multiply 15 and 6.3 to get 94.5.

$$116x=x^{2}+94.5+10\times 30.8$$

Multiply 10 and 30.8 to get 308.

$$116x=x^{2}+94.5+308$$

Add 94.5 and 308 to get 402.5.

$$116x=x^{2}+402.5$$

Subtract x^{2} from both sides.

$$116x-x^{2}=402.5$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$-x^{2}+116x=402.5$$

Divide both sides by -1.

$$\frac{-x^{2}+116x}{-1}=\frac{402.5}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{116}{-1}x=\frac{402.5}{-1}$$

Divide 116 by -1.

$$x^{2}-116x=\frac{402.5}{-1}$$

Divide 402.5 by -1.

$$x^{2}-116x=-402.5$$

Divide -116, the coefficient of the x term, by 2 to get -58. Then add the square of -58 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-116x+\left(-58\right)^{2}=-402.5+\left(-58\right)^{2}$$

Square -58.

$$x^{2}-116x+3364=-402.5+3364$$

Add -402.5 to 3364.

$$x^{2}-116x+3364=2961.5$$

Factor x^{2}-116x+3364. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-58\right)^{2}=2961.5$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-58\right)^{2}}=\sqrt{2961.5}$$

Simplify.

$$x-58=\frac{\sqrt{11846}}{2} x-58=-\frac{\sqrt{11846}}{2}$$

Add 58 to both sides of the equation.

$$x=\frac{\sqrt{11846}}{2}+58 x=-\frac{\sqrt{11846}}{2}+58$$